Let $X$ be a random variable. Define $P(X = j ) = p_j$ and $P(X>j) = q_j$. Let $P(s) = p_0 + p_1s + p_2s^2 + \cdot\cdot\cdot$ and $Q(s) = q_0 + q_1s + q_2s^2+ \cdot\cdot\cdot$. That is, $P(s)$ and $Q(s)$ are generating functions of $\{p_j\}$ and $\{q_j\}$, respectively.
I don't understand the statement that "since both functions are monotone this implies that $P'(s)$ and $Q(s)$ have the same finite or infinite limit ... ". How does MVT and monotonicity imply that these two have the same limit? Could youelaborate this?
Thanks in advance.
